the radon-nikody ́m theorem for vector measures and factorization of operators on banach function...
TRANSCRIPT
![Page 1: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/1.jpg)
The Radon-Nikodym Theorem for vector measures andfactorization of operators on Banach function spaces
Enrique A. Sanchez Perez
I.U.M.P.A.-U. Politecnica de Valencia,Joint work with O. Delgado (Universidad de Sevilla)
CIDAMA. Almerıa, 12-16 de septiembre de 2011
E. Sanchez Factorization of operators on Banach function spaces
![Page 2: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/2.jpg)
(Ω,Σ,µ) finite measure space.
X (µ) Banach function space over µ (Banach ideal ofmeasurable functions).
T : X (µ)→ E linear and continuous operator, E Banachspace.
E. Sanchez Factorization of operators on Banach function spaces
![Page 3: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/3.jpg)
(Ω,Σ,µ) finite measure space.
X (µ) Banach function space over µ (Banach ideal ofmeasurable functions).
T : X (µ)→ E linear and continuous operator, E Banachspace.
E. Sanchez Factorization of operators on Banach function spaces
![Page 4: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/4.jpg)
(Ω,Σ,µ) finite measure space.
X (µ) Banach function space over µ (Banach ideal ofmeasurable functions).
T : X (µ)→ E linear and continuous operator, E Banachspace.
E. Sanchez Factorization of operators on Banach function spaces
![Page 5: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/5.jpg)
General Integral Representation of Operators
T : X(µ)→ E , mT (A) := T (χA), T (f ) =∫
fd mT .
Radon-Nikodym Theorem, Diestel, Uhl
T : L1(µ)→ E , T (f ) =∫
φ fd µ.
Radon-Nikodym Theorem for Vector Measures
m,n : Σ→ E , m n, m(A) =∫
Afdn.
E. Sanchez Factorization of operators on Banach function spaces
![Page 6: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/6.jpg)
General Integral Representation of Operators
T : X(µ)→ E , mT (A) := T (χA), T (f ) =∫
fd mT .
Radon-Nikodym Theorem, Diestel, Uhl
T : L1(µ)→ E , T (f ) =∫
φ fd µ.
Radon-Nikodym Theorem for Vector Measures
m,n : Σ→ E , m n, m(A) =∫
Afdn.
E. Sanchez Factorization of operators on Banach function spaces
![Page 7: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/7.jpg)
General Integral Representation of Operators
T : X(µ)→ E , mT (A) := T (χA), T (f ) =∫
fd mT .
Radon-Nikodym Theorem, Diestel, Uhl
T : L1(µ)→ E , T (f ) =∫
φ fd µ.
Radon-Nikodym Theorem for Vector Measures
m,n : Σ→ E , m n, m(A) =∫
Afdn.
E. Sanchez Factorization of operators on Banach function spaces
![Page 8: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/8.jpg)
NOTATION: Banach function spaces
(Ω,Σ,µ) be a finite measure space.
L0(µ) space of all (classes of) measurable real functions on Ω.
A Banach function space (briefly B.f.s.) is a Banach space X ⊂ L0(µ) with norm‖ · ‖X such that if f ∈ L0(µ), g ∈ X and |f | ≤ |g| µ-a.e. then f ∈ X and ‖f‖X ≤ ‖g‖X .
A B.f.s. X has the Fatou property if for every sequence (fn)⊂ X such that 0≤ fn ↑ fµ-a.e. and supn ‖fn‖X < ∞, it follows that f ∈ X and ‖fn‖X ↑ ‖f‖X .
We will say that X is order continuous if for every f , fn ∈ X such that 0≤ fn ↑ fµ-a.e., we have that fn→ f in norm.
E. Sanchez Factorization of operators on Banach function spaces
![Page 9: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/9.jpg)
NOTATION: Banach function spaces
(Ω,Σ,µ) be a finite measure space.
L0(µ) space of all (classes of) measurable real functions on Ω.
A Banach function space (briefly B.f.s.) is a Banach space X ⊂ L0(µ) with norm‖ · ‖X such that if f ∈ L0(µ), g ∈ X and |f | ≤ |g| µ-a.e. then f ∈ X and ‖f‖X ≤ ‖g‖X .
A B.f.s. X has the Fatou property if for every sequence (fn)⊂ X such that 0≤ fn ↑ fµ-a.e. and supn ‖fn‖X < ∞, it follows that f ∈ X and ‖fn‖X ↑ ‖f‖X .
We will say that X is order continuous if for every f , fn ∈ X such that 0≤ fn ↑ fµ-a.e., we have that fn→ f in norm.
E. Sanchez Factorization of operators on Banach function spaces
![Page 10: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/10.jpg)
NOTATION: Banach function spaces
(Ω,Σ,µ) be a finite measure space.
L0(µ) space of all (classes of) measurable real functions on Ω.
A Banach function space (briefly B.f.s.) is a Banach space X ⊂ L0(µ) with norm‖ · ‖X such that if f ∈ L0(µ), g ∈ X and |f | ≤ |g| µ-a.e. then f ∈ X and ‖f‖X ≤ ‖g‖X .
A B.f.s. X has the Fatou property if for every sequence (fn)⊂ X such that 0≤ fn ↑ fµ-a.e. and supn ‖fn‖X < ∞, it follows that f ∈ X and ‖fn‖X ↑ ‖f‖X .
We will say that X is order continuous if for every f , fn ∈ X such that 0≤ fn ↑ fµ-a.e., we have that fn→ f in norm.
E. Sanchez Factorization of operators on Banach function spaces
![Page 11: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/11.jpg)
NOTATION: Banach function spaces
(Ω,Σ,µ) be a finite measure space.
L0(µ) space of all (classes of) measurable real functions on Ω.
A Banach function space (briefly B.f.s.) is a Banach space X ⊂ L0(µ) with norm‖ · ‖X such that if f ∈ L0(µ), g ∈ X and |f | ≤ |g| µ-a.e. then f ∈ X and ‖f‖X ≤ ‖g‖X .
A B.f.s. X has the Fatou property if for every sequence (fn)⊂ X such that 0≤ fn ↑ fµ-a.e. and supn ‖fn‖X < ∞, it follows that f ∈ X and ‖fn‖X ↑ ‖f‖X .
We will say that X is order continuous if for every f , fn ∈ X such that 0≤ fn ↑ fµ-a.e., we have that fn→ f in norm.
E. Sanchez Factorization of operators on Banach function spaces
![Page 12: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/12.jpg)
NOTATION: Banach function spaces
(Ω,Σ,µ) be a finite measure space.
L0(µ) space of all (classes of) measurable real functions on Ω.
A Banach function space (briefly B.f.s.) is a Banach space X ⊂ L0(µ) with norm‖ · ‖X such that if f ∈ L0(µ), g ∈ X and |f | ≤ |g| µ-a.e. then f ∈ X and ‖f‖X ≤ ‖g‖X .
A B.f.s. X has the Fatou property if for every sequence (fn)⊂ X such that 0≤ fn ↑ fµ-a.e. and supn ‖fn‖X < ∞, it follows that f ∈ X and ‖fn‖X ↑ ‖f‖X .
We will say that X is order continuous if for every f , fn ∈ X such that 0≤ fn ↑ fµ-a.e., we have that fn→ f in norm.
E. Sanchez Factorization of operators on Banach function spaces
![Page 13: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/13.jpg)
NOTATION: Banach function spaces
(Ω,Σ,µ) be a finite measure space.
L0(µ) space of all (classes of) measurable real functions on Ω.
A Banach function space (briefly B.f.s.) is a Banach space X ⊂ L0(µ) with norm‖ · ‖X such that if f ∈ L0(µ), g ∈ X and |f | ≤ |g| µ-a.e. then f ∈ X and ‖f‖X ≤ ‖g‖X .
A B.f.s. X has the Fatou property if for every sequence (fn)⊂ X such that 0≤ fn ↑ fµ-a.e. and supn ‖fn‖X < ∞, it follows that f ∈ X and ‖fn‖X ↑ ‖f‖X .
We will say that X is order continuous if for every f , fn ∈ X such that 0≤ fn ↑ fµ-a.e., we have that fn→ f in norm.
E. Sanchez Factorization of operators on Banach function spaces
![Page 14: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/14.jpg)
NOTATION: Multiplication operators
Given two B.f.s.’ X and Y , the space of multipliers from X to Y is defined as
X Y =
h ∈ L0(µ) : hf ∈ Y for all f ∈ X.
Seminorm on X Y : ‖ ·‖XY given by ‖h‖XY = supf∈BX‖hf‖Y for all h ∈ X Y . Is a norm
only in the case when X is saturated, i.e. there is g ∈ X such that g > 0 µ-a.e.In this case, X Y is a B.f.s.
X L1is the Kothe dual of X (X ′). For each h ∈ X Y , we denote by Mh : X → Y the
multiplication operator defined as Mh(f ) = hf for all f ∈ X .
E. Sanchez Factorization of operators on Banach function spaces
![Page 15: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/15.jpg)
NOTATION: Multiplication operators
Given two B.f.s.’ X and Y , the space of multipliers from X to Y is defined as
X Y =
h ∈ L0(µ) : hf ∈ Y for all f ∈ X.
Seminorm on X Y : ‖ ·‖XY given by ‖h‖XY = supf∈BX‖hf‖Y for all h ∈ X Y . Is a norm
only in the case when X is saturated, i.e. there is g ∈ X such that g > 0 µ-a.e.In this case, X Y is a B.f.s.
X L1is the Kothe dual of X (X ′). For each h ∈ X Y , we denote by Mh : X → Y the
multiplication operator defined as Mh(f ) = hf for all f ∈ X .
E. Sanchez Factorization of operators on Banach function spaces
![Page 16: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/16.jpg)
NOTATION: Multiplication operators
Given two B.f.s.’ X and Y , the space of multipliers from X to Y is defined as
X Y =
h ∈ L0(µ) : hf ∈ Y for all f ∈ X.
Seminorm on X Y : ‖ ·‖XY given by ‖h‖XY = supf∈BX‖hf‖Y for all h ∈ X Y . Is a norm
only in the case when X is saturated, i.e. there is g ∈ X such that g > 0 µ-a.e.In this case, X Y is a B.f.s.
X L1is the Kothe dual of X (X ′). For each h ∈ X Y , we denote by Mh : X → Y the
multiplication operator defined as Mh(f ) = hf for all f ∈ X .
E. Sanchez Factorization of operators on Banach function spaces
![Page 17: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/17.jpg)
NOTATION: Multiplication operators
Given two B.f.s.’ X and Y , the space of multipliers from X to Y is defined as
X Y =
h ∈ L0(µ) : hf ∈ Y for all f ∈ X.
Seminorm on X Y : ‖ ·‖XY given by ‖h‖XY = supf∈BX‖hf‖Y for all h ∈ X Y . Is a norm
only in the case when X is saturated, i.e. there is g ∈ X such that g > 0 µ-a.e.In this case, X Y is a B.f.s.
X L1is the Kothe dual of X (X ′). For each h ∈ X Y , we denote by Mh : X → Y the
multiplication operator defined as Mh(f ) = hf for all f ∈ X .
E. Sanchez Factorization of operators on Banach function spaces
![Page 18: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/18.jpg)
NOTATION: Product spaces
The product space XπY of two B.f.s.’ X and Y is the space of functions f ∈ L0(µ)such that |f | ≤ ∑i≥1 |xi yi | µ-a.e. for some sequences (xi )⊂ X and (yi )⊂ Ysatisfying ∑i≥1 ‖xi‖X ‖yi‖Y < ∞.
For f ∈ XπY , denote‖f‖XπY = ınf
∑i≥1‖xi‖X ‖yi‖Y
,
where the infimum is taken over all sequences (xi )⊂ X and (yi )⊂ Y such that|f | ≤∑i≥1 |xi yi | µ-a.e. and ∑i≥1 ‖xi‖X ‖yi‖Y < ∞. If X , Y and X Y ′ are saturated thenXπY is a saturated B.f.s. with norm ‖ · ‖XπY .
Theorem
Let X and Y be two B.f.s.’ containing L∞(µ) such that X is order continuous, the simplefunctions are dense in Y and X Y ′ is saturated. Then XπY is order continuous.
E. Sanchez Factorization of operators on Banach function spaces
![Page 19: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/19.jpg)
NOTATION: Product spaces
The product space XπY of two B.f.s.’ X and Y is the space of functions f ∈ L0(µ)such that |f | ≤ ∑i≥1 |xi yi | µ-a.e. for some sequences (xi )⊂ X and (yi )⊂ Ysatisfying ∑i≥1 ‖xi‖X ‖yi‖Y < ∞.
For f ∈ XπY , denote‖f‖XπY = ınf
∑i≥1‖xi‖X ‖yi‖Y
,
where the infimum is taken over all sequences (xi )⊂ X and (yi )⊂ Y such that|f | ≤∑i≥1 |xi yi | µ-a.e. and ∑i≥1 ‖xi‖X ‖yi‖Y < ∞. If X , Y and X Y ′ are saturated thenXπY is a saturated B.f.s. with norm ‖ · ‖XπY .
Theorem
Let X and Y be two B.f.s.’ containing L∞(µ) such that X is order continuous, the simplefunctions are dense in Y and X Y ′ is saturated. Then XπY is order continuous.
E. Sanchez Factorization of operators on Banach function spaces
![Page 20: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/20.jpg)
NOTATION: Product spaces
The product space XπY of two B.f.s.’ X and Y is the space of functions f ∈ L0(µ)such that |f | ≤ ∑i≥1 |xi yi | µ-a.e. for some sequences (xi )⊂ X and (yi )⊂ Ysatisfying ∑i≥1 ‖xi‖X ‖yi‖Y < ∞.
For f ∈ XπY , denote‖f‖XπY = ınf
∑i≥1‖xi‖X ‖yi‖Y
,
where the infimum is taken over all sequences (xi )⊂ X and (yi )⊂ Y such that|f | ≤∑i≥1 |xi yi | µ-a.e. and ∑i≥1 ‖xi‖X ‖yi‖Y < ∞. If X , Y and X Y ′ are saturated thenXπY is a saturated B.f.s. with norm ‖ · ‖XπY .
Theorem
Let X and Y be two B.f.s.’ containing L∞(µ) such that X is order continuous, the simplefunctions are dense in Y and X Y ′ is saturated. Then XπY is order continuous.
E. Sanchez Factorization of operators on Banach function spaces
![Page 21: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/21.jpg)
NOTATION: Product spaces
The product space XπY of two B.f.s.’ X and Y is the space of functions f ∈ L0(µ)such that |f | ≤ ∑i≥1 |xi yi | µ-a.e. for some sequences (xi )⊂ X and (yi )⊂ Ysatisfying ∑i≥1 ‖xi‖X ‖yi‖Y < ∞.
For f ∈ XπY , denote‖f‖XπY = ınf
∑i≥1‖xi‖X ‖yi‖Y
,
where the infimum is taken over all sequences (xi )⊂ X and (yi )⊂ Y such that|f | ≤∑i≥1 |xi yi | µ-a.e. and ∑i≥1 ‖xi‖X ‖yi‖Y < ∞. If X , Y and X Y ′ are saturated thenXπY is a saturated B.f.s. with norm ‖ · ‖XπY .
Theorem
Let X and Y be two B.f.s.’ containing L∞(µ) such that X is order continuous, the simplefunctions are dense in Y and X Y ′ is saturated. Then XπY is order continuous.
E. Sanchez Factorization of operators on Banach function spaces
![Page 22: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/22.jpg)
NOTATION: Vector measures and integration
Let m : Σ→ E be a vector measure, that is, a countably additive set function,where E is a real Banach space.
A set A ∈Σ is m-null if m(B) = 0 for every B ∈Σ with B ⊂ A. For each x∗ in thetopological dual E∗ of E , we denote by |x∗m| the variation of the real measurex∗m given by the composition of m with x∗. There exists x∗0 ∈ E∗ such that |x∗0m|has the same null sets as m. We will call |x∗0m| a Rybakov control measure for m.
A measurable function f : Ω→ R is integrable with respect to m if(i)
∫|f |d |x∗m|< ∞ for all x∗ ∈ E∗.
(ii) For each A ∈Σ, there exists xA ∈ E such that
x∗(xA) =∫
Af dx∗m, for all x∗ ∈ E .
The element xA will be written as∫
A f dm.
E. Sanchez Factorization of operators on Banach function spaces
![Page 23: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/23.jpg)
NOTATION: Vector measures and integration
Let m : Σ→ E be a vector measure, that is, a countably additive set function,where E is a real Banach space.
A set A ∈Σ is m-null if m(B) = 0 for every B ∈Σ with B ⊂ A. For each x∗ in thetopological dual E∗ of E , we denote by |x∗m| the variation of the real measurex∗m given by the composition of m with x∗. There exists x∗0 ∈ E∗ such that |x∗0m|has the same null sets as m. We will call |x∗0m| a Rybakov control measure for m.
A measurable function f : Ω→ R is integrable with respect to m if(i)
∫|f |d |x∗m|< ∞ for all x∗ ∈ E∗.
(ii) For each A ∈Σ, there exists xA ∈ E such that
x∗(xA) =∫
Af dx∗m, for all x∗ ∈ E .
The element xA will be written as∫
A f dm.
E. Sanchez Factorization of operators on Banach function spaces
![Page 24: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/24.jpg)
NOTATION: Vector measures and integration
Let m : Σ→ E be a vector measure, that is, a countably additive set function,where E is a real Banach space.
A set A ∈Σ is m-null if m(B) = 0 for every B ∈Σ with B ⊂ A. For each x∗ in thetopological dual E∗ of E , we denote by |x∗m| the variation of the real measurex∗m given by the composition of m with x∗. There exists x∗0 ∈ E∗ such that |x∗0m|has the same null sets as m. We will call |x∗0m| a Rybakov control measure for m.
A measurable function f : Ω→ R is integrable with respect to m if(i)
∫|f |d |x∗m|< ∞ for all x∗ ∈ E∗.
(ii) For each A ∈Σ, there exists xA ∈ E such that
x∗(xA) =∫
Af dx∗m, for all x∗ ∈ E .
The element xA will be written as∫
A f dm.
E. Sanchez Factorization of operators on Banach function spaces
![Page 25: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/25.jpg)
NOTATION: Vector measures and integration
Let m : Σ→ E be a vector measure, that is, a countably additive set function,where E is a real Banach space.
A set A ∈Σ is m-null if m(B) = 0 for every B ∈Σ with B ⊂ A. For each x∗ in thetopological dual E∗ of E , we denote by |x∗m| the variation of the real measurex∗m given by the composition of m with x∗. There exists x∗0 ∈ E∗ such that |x∗0m|has the same null sets as m. We will call |x∗0m| a Rybakov control measure for m.
A measurable function f : Ω→ R is integrable with respect to m if(i)
∫|f |d |x∗m|< ∞ for all x∗ ∈ E∗.
(ii) For each A ∈Σ, there exists xA ∈ E such that
x∗(xA) =∫
Af dx∗m, for all x∗ ∈ E .
The element xA will be written as∫
A f dm.
E. Sanchez Factorization of operators on Banach function spaces
![Page 26: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/26.jpg)
NOTATION: Vector measures and integration
Let m : Σ→ E be a vector measure, that is, a countably additive set function,where E is a real Banach space.
A set A ∈Σ is m-null if m(B) = 0 for every B ∈Σ with B ⊂ A. For each x∗ in thetopological dual E∗ of E , we denote by |x∗m| the variation of the real measurex∗m given by the composition of m with x∗. There exists x∗0 ∈ E∗ such that |x∗0m|has the same null sets as m. We will call |x∗0m| a Rybakov control measure for m.
A measurable function f : Ω→ R is integrable with respect to m if(i)
∫|f |d |x∗m|< ∞ for all x∗ ∈ E∗.
(ii) For each A ∈Σ, there exists xA ∈ E such that
x∗(xA) =∫
Af dx∗m, for all x∗ ∈ E .
The element xA will be written as∫
A f dm.
E. Sanchez Factorization of operators on Banach function spaces
![Page 27: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/27.jpg)
NOTATION: Spaces of integrable functions
Denote by L1(m) the space of integrable functions with respect to m, wherefunctions which are equal m-a.e. are identified.
The space L1(m) is a Banach space endowed with the norm
‖f‖m = supx∗∈BE∗
∫|f |d |x∗m|.
Note that L∞(|x∗0m|)⊂ L1(m). In particular every measure of the type |x∗m| is finiteas |x∗m|(Ω)≤ ‖x∗‖ · ‖χΩ‖m.
Given f ∈ L1(m), the set function mf : Σ→ E given by mf (A) =∫
A f dm for all A ∈Σis a vector measure. Moreover, g ∈ L1(mf ) if and only if gf ∈ L1(m) and in thiscase
∫g dmf =
∫gf dm.
E. Sanchez Factorization of operators on Banach function spaces
![Page 28: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/28.jpg)
NOTATION: Spaces of integrable functions
Denote by L1(m) the space of integrable functions with respect to m, wherefunctions which are equal m-a.e. are identified.
The space L1(m) is a Banach space endowed with the norm
‖f‖m = supx∗∈BE∗
∫|f |d |x∗m|.
Note that L∞(|x∗0m|)⊂ L1(m). In particular every measure of the type |x∗m| is finiteas |x∗m|(Ω)≤ ‖x∗‖ · ‖χΩ‖m.
Given f ∈ L1(m), the set function mf : Σ→ E given by mf (A) =∫
A f dm for all A ∈Σis a vector measure. Moreover, g ∈ L1(mf ) if and only if gf ∈ L1(m) and in thiscase
∫g dmf =
∫gf dm.
E. Sanchez Factorization of operators on Banach function spaces
![Page 29: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/29.jpg)
NOTATION: Spaces of integrable functions
Denote by L1(m) the space of integrable functions with respect to m, wherefunctions which are equal m-a.e. are identified.
The space L1(m) is a Banach space endowed with the norm
‖f‖m = supx∗∈BE∗
∫|f |d |x∗m|.
Note that L∞(|x∗0m|)⊂ L1(m). In particular every measure of the type |x∗m| is finiteas |x∗m|(Ω)≤ ‖x∗‖ · ‖χΩ‖m.
Given f ∈ L1(m), the set function mf : Σ→ E given by mf (A) =∫
A f dm for all A ∈Σis a vector measure. Moreover, g ∈ L1(mf ) if and only if gf ∈ L1(m) and in thiscase
∫g dmf =
∫gf dm.
E. Sanchez Factorization of operators on Banach function spaces
![Page 30: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/30.jpg)
NOTATION: Spaces of integrable functions
Denote by L1(m) the space of integrable functions with respect to m, wherefunctions which are equal m-a.e. are identified.
The space L1(m) is a Banach space endowed with the norm
‖f‖m = supx∗∈BE∗
∫|f |d |x∗m|.
Note that L∞(|x∗0m|)⊂ L1(m). In particular every measure of the type |x∗m| is finiteas |x∗m|(Ω)≤ ‖x∗‖ · ‖χΩ‖m.
Given f ∈ L1(m), the set function mf : Σ→ E given by mf (A) =∫
A f dm for all A ∈Σis a vector measure. Moreover, g ∈ L1(mf ) if and only if gf ∈ L1(m) and in thiscase
∫g dmf =
∫gf dm.
E. Sanchez Factorization of operators on Banach function spaces
![Page 31: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/31.jpg)
Main technical tool: Hahn-Banach Theorem + Ky Fan Lemma.
T : X(µ)→ E , ∑ni=1 ‖T (xi )‖p ≤ ‖∑
ni=1 |xi |p‖.
Family F of function Φ : B(Xp)′ → R
Defined for x1, ...xn ∈ X by
Φ(x ′) :=n
∑i=1‖T (xi )‖p−
n
∑i=1
∫|xi ·x ′|p dµ
Hahn-Banach Theorem: For each Φ ∈F there exists x ′Φ ∈ B(Xp)′ , such thatΦ(x ′Φ)≤ 0.
⇒ There exists x ′0 ∈ B(Xp)′ such that Φ(x ′0)≤ 0 for all Φ ∈F .
E. Sanchez Factorization of operators on Banach function spaces
![Page 32: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/32.jpg)
Main technical tool: Hahn-Banach Theorem + Ky Fan Lemma.
T : X(µ)→ E , ∑ni=1 ‖T (xi )‖p ≤ ‖∑
ni=1 |xi |p‖.
Family F of function Φ : B(Xp)′ → R
Defined for x1, ...xn ∈ X by
Φ(x ′) :=n
∑i=1‖T (xi )‖p−
n
∑i=1
∫|xi ·x ′|p dµ
Hahn-Banach Theorem: For each Φ ∈F there exists x ′Φ ∈ B(Xp)′ , such thatΦ(x ′Φ)≤ 0.
⇒ There exists x ′0 ∈ B(Xp)′ such that Φ(x ′0)≤ 0 for all Φ ∈F .
E. Sanchez Factorization of operators on Banach function spaces
![Page 33: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/33.jpg)
Main technical tool: Hahn-Banach Theorem + Ky Fan Lemma.
T : X(µ)→ E , ∑ni=1 ‖T (xi )‖p ≤ ‖∑
ni=1 |xi |p‖.
Family F of function Φ : B(Xp)′ → R
Defined for x1, ...xn ∈ X by
Φ(x ′) :=n
∑i=1‖T (xi )‖p−
n
∑i=1
∫|xi ·x ′|p dµ
Hahn-Banach Theorem: For each Φ ∈F there exists x ′Φ ∈ B(Xp)′ , such thatΦ(x ′Φ)≤ 0.
⇒ There exists x ′0 ∈ B(Xp)′ such that Φ(x ′0)≤ 0 for all Φ ∈F .
E. Sanchez Factorization of operators on Banach function spaces
![Page 34: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/34.jpg)
Main technical tool: Hahn-Banach Theorem + Ky Fan Lemma.
T : X(µ)→ E , ∑ni=1 ‖T (xi )‖p ≤ ‖∑
ni=1 |xi |p‖.
Family F of function Φ : B(Xp)′ → R
Defined for x1, ...xn ∈ X by
Φ(x ′) :=n
∑i=1‖T (xi )‖p−
n
∑i=1
∫|xi ·x ′|p dµ
Hahn-Banach Theorem: For each Φ ∈F there exists x ′Φ ∈ B(Xp)′ , such thatΦ(x ′Φ)≤ 0.
⇒ There exists x ′0 ∈ B(Xp)′ such that Φ(x ′0)≤ 0 for all Φ ∈F .
E. Sanchez Factorization of operators on Banach function spaces
![Page 35: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/35.jpg)
Main technical tool: Hahn-Banach Theorem + Ky Fan Lemma.
T : X(µ)→ E , ∑ni=1 ‖T (xi )‖p ≤ ‖∑
ni=1 |xi |p‖.
Family F of function Φ : B(Xp)′ → R
Defined for x1, ...xn ∈ X by
Φ(x ′) :=n
∑i=1‖T (xi )‖p−
n
∑i=1
∫|xi ·x ′|p dµ
Hahn-Banach Theorem: For each Φ ∈F there exists x ′Φ ∈ B(Xp)′ , such thatΦ(x ′Φ)≤ 0.
⇒ There exists x ′0 ∈ B(Xp)′ such that Φ(x ′0)≤ 0 for all Φ ∈F .
E. Sanchez Factorization of operators on Banach function spaces
![Page 36: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/36.jpg)
Main technical tool: Hahn-Banach Theorem + Ky Fan Lemma.
T : X(µ)→ E , ∑ni=1 ‖T (xi )‖p ≤ ‖∑
ni=1 |xi |p‖.
Family F of function Φ : B(Xp)′ → R
Defined for x1, ...xn ∈ X by
Φ(x ′) :=n
∑i=1‖T (xi )‖p−
n
∑i=1
∫|xi ·x ′|p dµ
Hahn-Banach Theorem: For each Φ ∈F there exists x ′Φ ∈ B(Xp)′ , such thatΦ(x ′Φ)≤ 0.
⇒ There exists x ′0 ∈ B(Xp)′ such that Φ(x ′0)≤ 0 for all Φ ∈F .
E. Sanchez Factorization of operators on Banach function spaces
![Page 37: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/37.jpg)
THE TALK:
1. A Radon-Nikodym Theorem for vector measures.
2. Strong factorization of operators on Banach function spaces.
3. Some classical and new applications.
E. Sanchez Factorization of operators on Banach function spaces
![Page 38: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/38.jpg)
THE TALK:
1. A Radon-Nikodym Theorem for vector measures.
2. Strong factorization of operators on Banach function spaces.
3. Some classical and new applications.
E. Sanchez Factorization of operators on Banach function spaces
![Page 39: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/39.jpg)
THE TALK:
1. A Radon-Nikodym Theorem for vector measures.
2. Strong factorization of operators on Banach function spaces.
3. Some classical and new applications.
E. Sanchez Factorization of operators on Banach function spaces
![Page 40: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/40.jpg)
THE TALK:
1. A Radon-Nikodym Theorem for vector measures.
2. Strong factorization of operators on Banach function spaces.
3. Some classical and new applications.
E. Sanchez Factorization of operators on Banach function spaces
![Page 41: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/41.jpg)
1. A Radon-Nikodym Theorem for vector measures.
E. Sanchez Factorization of operators on Banach function spaces
![Page 42: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/42.jpg)
K. Musial, A Radon-Nikodym theorem for the Bartle-Dunford-Schwartz integral, AttiSem. Mat. Fis. Univ. Modena XLI (1993), 227-233.
Theorem
Let m,n : Σ→ E be two vector measures and take |x∗0m| a Rybakov control measurefor m. The following statements are equivalent.
(1) There exists a positive constant K such that
x∗n(A)≤ K |x∗m|(A) for all A ∈Σ and x∗ ∈ E∗.
(2) There exists a function g ∈ L∞(|x∗0m|) such that
n(A) =∫
Ag dm for all A ∈Σ.
E. Sanchez Factorization of operators on Banach function spaces
![Page 43: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/43.jpg)
K. Musial, A Radon-Nikodym theorem for the Bartle-Dunford-Schwartz integral, AttiSem. Mat. Fis. Univ. Modena XLI (1993), 227-233.
Theorem
Let m,n : Σ→ E be two vector measures and take |x∗0m| a Rybakov control measurefor m. The following statements are equivalent.
(1) There exists a positive constant K such that
x∗n(A)≤ K |x∗m|(A) for all A ∈Σ and x∗ ∈ E∗.
(2) There exists a function g ∈ L∞(|x∗0m|) such that
n(A) =∫
Ag dm for all A ∈Σ.
E. Sanchez Factorization of operators on Banach function spaces
![Page 44: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/44.jpg)
2. Strong factorization of operators on Banach function spaces.
E. Sanchez Factorization of operators on Banach function spaces
![Page 45: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/45.jpg)
Let (Ω,Σ,µ) be a fixed finite measure space and X1,X2,Y1,Y2 B.f.s.’ related to µ
such that L∞(µ)⊂ X1 ⊂ X2 and L∞(µ)⊂ Y2 ⊂ Y1. This guarantees that X X21 and
Y Y12 are B.f.s.’ containing L∞(µ).
Consider two continuous linear operators T : X1→ Y1 and S : X2→ Y2.
When T factorizes strongly through S?
When the following diagram commutes
X1T //
Mf
Y1
X2S // Y2
Mg
OO (1)
for some f ∈ X X21 and g ∈ Y Y1
2 ?
E. Sanchez Factorization of operators on Banach function spaces
![Page 46: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/46.jpg)
Let (Ω,Σ,µ) be a fixed finite measure space and X1,X2,Y1,Y2 B.f.s.’ related to µ
such that L∞(µ)⊂ X1 ⊂ X2 and L∞(µ)⊂ Y2 ⊂ Y1. This guarantees that X X21 and
Y Y12 are B.f.s.’ containing L∞(µ).
Consider two continuous linear operators T : X1→ Y1 and S : X2→ Y2.
When T factorizes strongly through S?
When the following diagram commutes
X1T //
Mf
Y1
X2S // Y2
Mg
OO (1)
for some f ∈ X X21 and g ∈ Y Y1
2 ?
E. Sanchez Factorization of operators on Banach function spaces
![Page 47: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/47.jpg)
Let (Ω,Σ,µ) be a fixed finite measure space and X1,X2,Y1,Y2 B.f.s.’ related to µ
such that L∞(µ)⊂ X1 ⊂ X2 and L∞(µ)⊂ Y2 ⊂ Y1. This guarantees that X X21 and
Y Y12 are B.f.s.’ containing L∞(µ).
Consider two continuous linear operators T : X1→ Y1 and S : X2→ Y2.
When T factorizes strongly through S?
When the following diagram commutes
X1T //
Mf
Y1
X2S // Y2
Mg
OO (1)
for some f ∈ X X21 and g ∈ Y Y1
2 ?
E. Sanchez Factorization of operators on Banach function spaces
![Page 48: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/48.jpg)
Let (Ω,Σ,µ) be a fixed finite measure space and X1,X2,Y1,Y2 B.f.s.’ related to µ
such that L∞(µ)⊂ X1 ⊂ X2 and L∞(µ)⊂ Y2 ⊂ Y1. This guarantees that X X21 and
Y Y12 are B.f.s.’ containing L∞(µ).
Consider two continuous linear operators T : X1→ Y1 and S : X2→ Y2.
When T factorizes strongly through S?
When the following diagram commutes
X1T //
Mf
Y1
X2S // Y2
Mg
OO (1)
for some f ∈ X X21 and g ∈ Y Y1
2 ?
E. Sanchez Factorization of operators on Banach function spaces
![Page 49: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/49.jpg)
Let (Ω,Σ,µ) be a fixed finite measure space and X1,X2,Y1,Y2 B.f.s.’ related to µ
such that L∞(µ)⊂ X1 ⊂ X2 and L∞(µ)⊂ Y2 ⊂ Y1. This guarantees that X X21 and
Y Y12 are B.f.s.’ containing L∞(µ).
Consider two continuous linear operators T : X1→ Y1 and S : X2→ Y2.
When T factorizes strongly through S?
When the following diagram commutes
X1T //
Mf
Y1
X2S // Y2
Mg
OO (1)
for some f ∈ X X21 and g ∈ Y Y1
2 ?
E. Sanchez Factorization of operators on Banach function spaces
![Page 50: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/50.jpg)
Let (Ω,Σ,µ) be a fixed finite measure space and X1,X2,Y1,Y2 B.f.s.’ related to µ
such that L∞(µ)⊂ X1 ⊂ X2 and L∞(µ)⊂ Y2 ⊂ Y1. This guarantees that X X21 and
Y Y12 are B.f.s.’ containing L∞(µ).
Consider two continuous linear operators T : X1→ Y1 and S : X2→ Y2.
When T factorizes strongly through S?
When the following diagram commutes
X1T //
Mf
Y1
X2S // Y2
Mg
OO (1)
for some f ∈ X X21 and g ∈ Y Y1
2 ?
E. Sanchez Factorization of operators on Banach function spaces
![Page 51: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/51.jpg)
Theorem
Suppose that Y1,Y2πY ′1 are order continuous and moreover Y1 has the Fatou property.The following statements are equivalent:
(i) There exists a function h ∈ X X21 such that
n
∑i=1
∫T (xi )y ′i dµ ≤
∥∥∥ n
∑i=1
S(hxi )y ′i∥∥∥
Y2πY ′1
for every x1, ...,xn ∈ X1 and y ′1, ...,y′n ∈ Y ′1.
(ii) There exist functions f ∈ X X21 and g ∈ Y Y1
2 such that T (x) = gS(fx) for all x ∈ X1,i.e. T factorizes strongly through S as
X1T //
Mf
Y1
X2S // Y2
Mg
OO (2)
E. Sanchez Factorization of operators on Banach function spaces
![Page 52: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/52.jpg)
Theorem
Suppose that Y1,Y2πY ′1 are order continuous and moreover Y1 has the Fatou property.The following statements are equivalent:
(i) There exists a function h ∈ X X21 such that
n
∑i=1
∫T (xi )y ′i dµ ≤
∥∥∥ n
∑i=1
S(hxi )y ′i∥∥∥
Y2πY ′1
for every x1, ...,xn ∈ X1 and y ′1, ...,y′n ∈ Y ′1.
(ii) There exist functions f ∈ X X21 and g ∈ Y Y1
2 such that T (x) = gS(fx) for all x ∈ X1,i.e. T factorizes strongly through S as
X1T //
Mf
Y1
X2S // Y2
Mg
OO (2)
E. Sanchez Factorization of operators on Banach function spaces
![Page 53: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/53.jpg)
Theorem
Suppose that X1, Y1, X2 and X1πX ′2 are order continuous and X2 has the Fatouproperty. The following statements are equivalent.
(i) There exists a function h ∈ Y Y12 such that
n
∑i=1
∫T (xi )y ′i dµ ≤
∥∥∥ n
∑i=1|S′(hy ′i )xi |
∥∥∥X1πX ′2
for every x1, ...,xn ∈ X1 and y ′1, ...,y′n ∈ Y ′1.
(ii) There exist functions f ∈ X X21 and g ∈ Y Y1
2 such that T (x) = gS(fx) for all x ∈ X1,i.e. it factorizes as
X1T //
Mf
Y1
X2S // Y2
Mg
OO (3)
E. Sanchez Factorization of operators on Banach function spaces
![Page 54: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/54.jpg)
Theorem
Suppose that X1, Y1, X2 and X1πX ′2 are order continuous and X2 has the Fatouproperty. The following statements are equivalent.
(i) There exists a function h ∈ Y Y12 such that
n
∑i=1
∫T (xi )y ′i dµ ≤
∥∥∥ n
∑i=1|S′(hy ′i )xi |
∥∥∥X1πX ′2
for every x1, ...,xn ∈ X1 and y ′1, ...,y′n ∈ Y ′1.
(ii) There exist functions f ∈ X X21 and g ∈ Y Y1
2 such that T (x) = gS(fx) for all x ∈ X1,i.e. it factorizes as
X1T //
Mf
Y1
X2S // Y2
Mg
OO (3)
E. Sanchez Factorization of operators on Banach function spaces
![Page 55: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/55.jpg)
Example
If all X1,X2,Y1,Y2 coincide with an order continuous B.f.s. X having the Fatouproperty and containing L∞(µ).
Then XπX ′ = L1(µ) (with equal norms), that is order continuous (Lozanovskii).
Also X X = L∞(µ) (with equal norms). By the theorems given before we obtain:
E. Sanchez Factorization of operators on Banach function spaces
![Page 56: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/56.jpg)
Example
If all X1,X2,Y1,Y2 coincide with an order continuous B.f.s. X having the Fatouproperty and containing L∞(µ).
Then XπX ′ = L1(µ) (with equal norms), that is order continuous (Lozanovskii).
Also X X = L∞(µ) (with equal norms). By the theorems given before we obtain:
E. Sanchez Factorization of operators on Banach function spaces
![Page 57: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/57.jpg)
Example
If all X1,X2,Y1,Y2 coincide with an order continuous B.f.s. X having the Fatouproperty and containing L∞(µ).
Then XπX ′ = L1(µ) (with equal norms), that is order continuous (Lozanovskii).
Also X X = L∞(µ) (with equal norms). By the theorems given before we obtain:
E. Sanchez Factorization of operators on Banach function spaces
![Page 58: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/58.jpg)
Corollary
The following statements are equivalent:
(i) T factorizes strongly through S, i.e.
XT //
Mf
X
XS // X
Mg
OO
for some f ,g ∈ L∞(µ).
(ii) There exists a function h ∈ L∞(µ) such that
n
∑i=1
∫T (xi )x ′i dµ ≤
∫ ∣∣∣ n
∑i=1
S(hxi )x ′i∣∣∣dµ
for every x1, ...,xn ∈ X and x ′1, ...,x′n ∈ X ′.
(iii) There exists a function h ∈ L∞(µ) such that∫T (x)x ′ dµ ≤
∫|S′(hx ′)x |dµ
for every x ∈ X and x ′ ∈ X ′.
E. Sanchez Factorization of operators on Banach function spaces
![Page 59: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/59.jpg)
Corollary
The following statements are equivalent:
(i) T factorizes strongly through S, i.e.
XT //
Mf
X
XS // X
Mg
OO
for some f ,g ∈ L∞(µ).
(ii) There exists a function h ∈ L∞(µ) such that
n
∑i=1
∫T (xi )x ′i dµ ≤
∫ ∣∣∣ n
∑i=1
S(hxi )x ′i∣∣∣dµ
for every x1, ...,xn ∈ X and x ′1, ...,x′n ∈ X ′.
(iii) There exists a function h ∈ L∞(µ) such that∫T (x)x ′ dµ ≤
∫|S′(hx ′)x |dµ
for every x ∈ X and x ′ ∈ X ′.
E. Sanchez Factorization of operators on Banach function spaces
![Page 60: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/60.jpg)
Corollary
The following statements are equivalent:
(i) T factorizes strongly through S, i.e.
XT //
Mf
X
XS // X
Mg
OO
for some f ,g ∈ L∞(µ).
(ii) There exists a function h ∈ L∞(µ) such that
n
∑i=1
∫T (xi )x ′i dµ ≤
∫ ∣∣∣ n
∑i=1
S(hxi )x ′i∣∣∣dµ
for every x1, ...,xn ∈ X and x ′1, ...,x′n ∈ X ′.
(iii) There exists a function h ∈ L∞(µ) such that∫T (x)x ′ dµ ≤
∫|S′(hx ′)x |dµ
for every x ∈ X and x ′ ∈ X ′.
E. Sanchez Factorization of operators on Banach function spaces
![Page 61: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/61.jpg)
3. Some classical and new applications.
E. Sanchez Factorization of operators on Banach function spaces
![Page 62: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/62.jpg)
Maurey-Rosenthal type factorization of operators
Let X(µ) be an order continuous p-convex Banach function space, i.e.
‖(∑ |fi |p)1/p‖ ≤ K (∑‖fi‖p)1/p .
An operator T : X(µ)→ E factorizes strongly through Lp(µ) if and only if T is
p-concave, i.e.(∑‖T (fi )‖p)1/p ≤Q‖(∑ |fi |p)1/p‖.
Consider the following factorization diagram.
X(µ)T //
Mf
Y (µ)
Lp(µ)S // Y (µ)
Id
OO
and apply our arguments.
E. Sanchez Factorization of operators on Banach function spaces
![Page 63: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/63.jpg)
Maurey-Rosenthal type factorization of operators
Let X(µ) be an order continuous p-convex Banach function space, i.e.
‖(∑ |fi |p)1/p‖ ≤ K (∑‖fi‖p)1/p .
An operator T : X(µ)→ E factorizes strongly through Lp(µ) if and only if T is
p-concave, i.e.(∑‖T (fi )‖p)1/p ≤Q‖(∑ |fi |p)1/p‖.
Consider the following factorization diagram.
X(µ)T //
Mf
Y (µ)
Lp(µ)S // Y (µ)
Id
OO
and apply our arguments.
E. Sanchez Factorization of operators on Banach function spaces
![Page 64: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/64.jpg)
Maurey-Rosenthal type factorization of operators
Let X(µ) be an order continuous p-convex Banach function space, i.e.
‖(∑ |fi |p)1/p‖ ≤ K (∑‖fi‖p)1/p .
An operator T : X(µ)→ E factorizes strongly through Lp(µ) if and only if T is
p-concave, i.e.(∑‖T (fi )‖p)1/p ≤Q‖(∑ |fi |p)1/p‖.
Consider the following factorization diagram.
X(µ)T //
Mf
Y (µ)
Lp(µ)S // Y (µ)
Id
OO
and apply our arguments.
E. Sanchez Factorization of operators on Banach function spaces
![Page 65: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/65.jpg)
Maurey-Rosenthal type factorization of operators
Let X(µ) be an order continuous p-convex Banach function space, i.e.
‖(∑ |fi |p)1/p‖ ≤ K (∑‖fi‖p)1/p .
An operator T : X(µ)→ E factorizes strongly through Lp(µ) if and only if T is
p-concave, i.e.(∑‖T (fi )‖p)1/p ≤Q‖(∑ |fi |p)1/p‖.
Consider the following factorization diagram.
X(µ)T //
Mf
Y (µ)
Lp(µ)S // Y (µ)
Id
OO
and apply our arguments.
E. Sanchez Factorization of operators on Banach function spaces
![Page 66: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/66.jpg)
Strong factorization through a kernel operator
Consider a measurable function K : Ω×Ω→ [0,∞) such that the operator SK : X2→ Y2given by
SK (f )(s) =∫
f (t)K (s, t)dµ(t)
for all f ∈ X2 and s ∈Ω, is well defined and so continuous.
Under the requirements of the theorems, the statement:there exists a function h ∈ Y Y1
2 such that
n
∑i=1
∫T (xi )y ′i dµ ≤
∥∥∥ n
∑i=1|(SK )′(hy ′i )xi |
∥∥∥X1πX ′2
for every x1, ...,xn ∈ X1 and y ′1, ...,y′n ∈ Y ′1...
... holds if and only if T factorizes strongly through SK , that is, there exist f ∈ X X21
and g ∈ Y Y12 such that
T (x)(s) = g(s)∫
f (t)x(t)K (s, t)dµ(t)
for all x ∈ X1 and s ∈Ω. In this case, T is also a kernel operator with kernelK (s, t) = g(s)f (t)K (s, t).
E. Sanchez Factorization of operators on Banach function spaces
![Page 67: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/67.jpg)
Strong factorization through a kernel operator
Consider a measurable function K : Ω×Ω→ [0,∞) such that the operator SK : X2→ Y2given by
SK (f )(s) =∫
f (t)K (s, t)dµ(t)
for all f ∈ X2 and s ∈Ω, is well defined and so continuous.
Under the requirements of the theorems, the statement:there exists a function h ∈ Y Y1
2 such that
n
∑i=1
∫T (xi )y ′i dµ ≤
∥∥∥ n
∑i=1|(SK )′(hy ′i )xi |
∥∥∥X1πX ′2
for every x1, ...,xn ∈ X1 and y ′1, ...,y′n ∈ Y ′1...
... holds if and only if T factorizes strongly through SK , that is, there exist f ∈ X X21
and g ∈ Y Y12 such that
T (x)(s) = g(s)∫
f (t)x(t)K (s, t)dµ(t)
for all x ∈ X1 and s ∈Ω. In this case, T is also a kernel operator with kernelK (s, t) = g(s)f (t)K (s, t).
E. Sanchez Factorization of operators on Banach function spaces
![Page 68: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/68.jpg)
Strong factorization through a kernel operator
Consider a measurable function K : Ω×Ω→ [0,∞) such that the operator SK : X2→ Y2given by
SK (f )(s) =∫
f (t)K (s, t)dµ(t)
for all f ∈ X2 and s ∈Ω, is well defined and so continuous.
Under the requirements of the theorems, the statement:there exists a function h ∈ Y Y1
2 such that
n
∑i=1
∫T (xi )y ′i dµ ≤
∥∥∥ n
∑i=1|(SK )′(hy ′i )xi |
∥∥∥X1πX ′2
for every x1, ...,xn ∈ X1 and y ′1, ...,y′n ∈ Y ′1...
... holds if and only if T factorizes strongly through SK , that is, there exist f ∈ X X21
and g ∈ Y Y12 such that
T (x)(s) = g(s)∫
f (t)x(t)K (s, t)dµ(t)
for all x ∈ X1 and s ∈Ω. In this case, T is also a kernel operator with kernelK (s, t) = g(s)f (t)K (s, t).
E. Sanchez Factorization of operators on Banach function spaces
![Page 69: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/69.jpg)
Strong factorization through a kernel operator
Consider a measurable function K : Ω×Ω→ [0,∞) such that the operator SK : X2→ Y2given by
SK (f )(s) =∫
f (t)K (s, t)dµ(t)
for all f ∈ X2 and s ∈Ω, is well defined and so continuous.
Under the requirements of the theorems, the statement:there exists a function h ∈ Y Y1
2 such that
n
∑i=1
∫T (xi )y ′i dµ ≤
∥∥∥ n
∑i=1|(SK )′(hy ′i )xi |
∥∥∥X1πX ′2
for every x1, ...,xn ∈ X1 and y ′1, ...,y′n ∈ Y ′1...
... holds if and only if T factorizes strongly through SK , that is, there exist f ∈ X X21
and g ∈ Y Y12 such that
T (x)(s) = g(s)∫
f (t)x(t)K (s, t)dµ(t)
for all x ∈ X1 and s ∈Ω. In this case, T is also a kernel operator with kernelK (s, t) = g(s)f (t)K (s, t).
E. Sanchez Factorization of operators on Banach function spaces
![Page 70: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/70.jpg)
Strong factorization through a kernel operator
Consider a measurable function K : Ω×Ω→ [0,∞) such that the operator SK : X2→ Y2given by
SK (f )(s) =∫
f (t)K (s, t)dµ(t)
for all f ∈ X2 and s ∈Ω, is well defined and so continuous.
Under the requirements of the theorems, the statement:there exists a function h ∈ Y Y1
2 such that
n
∑i=1
∫T (xi )y ′i dµ ≤
∥∥∥ n
∑i=1|(SK )′(hy ′i )xi |
∥∥∥X1πX ′2
for every x1, ...,xn ∈ X1 and y ′1, ...,y′n ∈ Y ′1...
... holds if and only if T factorizes strongly through SK , that is, there exist f ∈ X X21
and g ∈ Y Y12 such that
T (x)(s) = g(s)∫
f (t)x(t)K (s, t)dµ(t)
for all x ∈ X1 and s ∈Ω. In this case, T is also a kernel operator with kernelK (s, t) = g(s)f (t)K (s, t).
E. Sanchez Factorization of operators on Banach function spaces
![Page 71: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/71.jpg)
Strong factorization through a kernel operator
Consider a measurable function K : Ω×Ω→ [0,∞) such that the operator SK : X2→ Y2given by
SK (f )(s) =∫
f (t)K (s, t)dµ(t)
for all f ∈ X2 and s ∈Ω, is well defined and so continuous.
Under the requirements of the theorems, the statement:there exists a function h ∈ Y Y1
2 such that
n
∑i=1
∫T (xi )y ′i dµ ≤
∥∥∥ n
∑i=1|(SK )′(hy ′i )xi |
∥∥∥X1πX ′2
for every x1, ...,xn ∈ X1 and y ′1, ...,y′n ∈ Y ′1...
... holds if and only if T factorizes strongly through SK , that is, there exist f ∈ X X21
and g ∈ Y Y12 such that
T (x)(s) = g(s)∫
f (t)x(t)K (s, t)dµ(t)
for all x ∈ X1 and s ∈Ω. In this case, T is also a kernel operator with kernelK (s, t) = g(s)f (t)K (s, t).
E. Sanchez Factorization of operators on Banach function spaces
![Page 72: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/72.jpg)
A particular case
If X1,X2,Y1,Y2 all coincide with an order continuous B.f.s. X having the Fatou propertyand containing L∞(µ), by the corollary, we obtain:
T factorizes strongly through SK if and only if ...
... there exists h ∈ L∞(µ) such that∫T (x)(t)x ′(t)dµ(t)≤
∫ ∣∣∣x(t)∫
h(s)x ′(s)K (s, t)dµ(s)∣∣∣dµ(t) (4)
for all x ∈ X and x ′ ∈ X ′.
Indeed, S′K : X ′→ X ′ satisfies that
〈S′K (x ′),x〉 = 〈x ′,SK (x)〉=∫
x ′(s)∫
x(t)K (s, t)dµ(t) dµ(s)
=∫
x(t)∫
x ′(s)K (s, t)dµ(s) dµ(t)
=⟨∫
x ′(s)K (s, ·)dµ(s),x⟩
for all x ′ ∈ X ′ and x ∈ X .
E. Sanchez Factorization of operators on Banach function spaces
![Page 73: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/73.jpg)
A particular case
If X1,X2,Y1,Y2 all coincide with an order continuous B.f.s. X having the Fatou propertyand containing L∞(µ), by the corollary, we obtain:
T factorizes strongly through SK if and only if ...
... there exists h ∈ L∞(µ) such that∫T (x)(t)x ′(t)dµ(t)≤
∫ ∣∣∣x(t)∫
h(s)x ′(s)K (s, t)dµ(s)∣∣∣dµ(t) (4)
for all x ∈ X and x ′ ∈ X ′.
Indeed, S′K : X ′→ X ′ satisfies that
〈S′K (x ′),x〉 = 〈x ′,SK (x)〉=∫
x ′(s)∫
x(t)K (s, t)dµ(t) dµ(s)
=∫
x(t)∫
x ′(s)K (s, t)dµ(s) dµ(t)
=⟨∫
x ′(s)K (s, ·)dµ(s),x⟩
for all x ′ ∈ X ′ and x ∈ X .
E. Sanchez Factorization of operators on Banach function spaces
![Page 74: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/74.jpg)
A particular case
If X1,X2,Y1,Y2 all coincide with an order continuous B.f.s. X having the Fatou propertyand containing L∞(µ), by the corollary, we obtain:
T factorizes strongly through SK if and only if ...
... there exists h ∈ L∞(µ) such that∫T (x)(t)x ′(t)dµ(t)≤
∫ ∣∣∣x(t)∫
h(s)x ′(s)K (s, t)dµ(s)∣∣∣dµ(t) (4)
for all x ∈ X and x ′ ∈ X ′.
Indeed, S′K : X ′→ X ′ satisfies that
〈S′K (x ′),x〉 = 〈x ′,SK (x)〉=∫
x ′(s)∫
x(t)K (s, t)dµ(t) dµ(s)
=∫
x(t)∫
x ′(s)K (s, t)dµ(s) dµ(t)
=⟨∫
x ′(s)K (s, ·)dµ(s),x⟩
for all x ′ ∈ X ′ and x ∈ X .
E. Sanchez Factorization of operators on Banach function spaces
![Page 75: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/75.jpg)
A particular case
If X1,X2,Y1,Y2 all coincide with an order continuous B.f.s. X having the Fatou propertyand containing L∞(µ), by the corollary, we obtain:
T factorizes strongly through SK if and only if ...
... there exists h ∈ L∞(µ) such that∫T (x)(t)x ′(t)dµ(t)≤
∫ ∣∣∣x(t)∫
h(s)x ′(s)K (s, t)dµ(s)∣∣∣dµ(t) (4)
for all x ∈ X and x ′ ∈ X ′.
Indeed, S′K : X ′→ X ′ satisfies that
〈S′K (x ′),x〉 = 〈x ′,SK (x)〉=∫
x ′(s)∫
x(t)K (s, t)dµ(t) dµ(s)
=∫
x(t)∫
x ′(s)K (s, t)dµ(s) dµ(t)
=⟨∫
x ′(s)K (s, ·)dµ(s),x⟩
for all x ′ ∈ X ′ and x ∈ X .
E. Sanchez Factorization of operators on Banach function spaces
![Page 76: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/76.jpg)
A particular case
If X1,X2,Y1,Y2 all coincide with an order continuous B.f.s. X having the Fatou propertyand containing L∞(µ), by the corollary, we obtain:
T factorizes strongly through SK if and only if ...
... there exists h ∈ L∞(µ) such that∫T (x)(t)x ′(t)dµ(t)≤
∫ ∣∣∣x(t)∫
h(s)x ′(s)K (s, t)dµ(s)∣∣∣dµ(t) (4)
for all x ∈ X and x ′ ∈ X ′.
Indeed, S′K : X ′→ X ′ satisfies that
〈S′K (x ′),x〉 = 〈x ′,SK (x)〉=∫
x ′(s)∫
x(t)K (s, t)dµ(t) dµ(s)
=∫
x(t)∫
x ′(s)K (s, t)dµ(s) dµ(t)
=⟨∫
x ′(s)K (s, ·)dµ(s),x⟩
for all x ′ ∈ X ′ and x ∈ X .
E. Sanchez Factorization of operators on Banach function spaces
![Page 77: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/77.jpg)
Example
Consider the measure space given by the interval [0,1], its Borel σ -algebra and theLebesgue measure. Let K be the kernel given by K (s, t) = χ[0,s](t) for all s, t ∈ [0,1].Then, SK is just the Volterra operator. Suppose that SK : X → X is well defined andcontinuous (e.g. X = Lp[0,1] with 1≤ p < ∞). The following statements are equivalent:
(i) There exist g, f ∈ L∞[0,1] such that
T (x)(s) = g(s)∫ s
0f (t)x(t)dt a.e.(s)
for all x ∈ X .
(ii) There exists h ∈ L∞[0,1] such that∫ 1
0T (x)(t)x ′(t)dt ≤
∫ 1
0
∣∣∣x(t)∫ 1
th(s)x ′(s)ds
∣∣∣dt
for all x ∈ X and x ′ ∈ X ′.
(iii) There exists h ∈ L∞[0,1] such that
|T ′(x ′)(t)| ≤∣∣∣∫ 1
th(s)x ′(s)ds
∣∣∣ a.e.(t)
for all x ′ ∈ X ′. E. Sanchez Factorization of operators on Banach function spaces
![Page 78: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/78.jpg)
Example
Consider the measure space given by the interval [0,1], its Borel σ -algebra and theLebesgue measure. Let K be the kernel given by K (s, t) = χ[0,s](t) for all s, t ∈ [0,1].Then, SK is just the Volterra operator. Suppose that SK : X → X is well defined andcontinuous (e.g. X = Lp[0,1] with 1≤ p < ∞). The following statements are equivalent:
(i) There exist g, f ∈ L∞[0,1] such that
T (x)(s) = g(s)∫ s
0f (t)x(t)dt a.e.(s)
for all x ∈ X .
(ii) There exists h ∈ L∞[0,1] such that∫ 1
0T (x)(t)x ′(t)dt ≤
∫ 1
0
∣∣∣x(t)∫ 1
th(s)x ′(s)ds
∣∣∣dt
for all x ∈ X and x ′ ∈ X ′.
(iii) There exists h ∈ L∞[0,1] such that
|T ′(x ′)(t)| ≤∣∣∣∫ 1
th(s)x ′(s)ds
∣∣∣ a.e.(t)
for all x ′ ∈ X ′. E. Sanchez Factorization of operators on Banach function spaces
![Page 79: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/79.jpg)
Example
Consider the measure space given by the interval [0,1], its Borel σ -algebra and theLebesgue measure. Let K be the kernel given by K (s, t) = χ[0,s](t) for all s, t ∈ [0,1].Then, SK is just the Volterra operator. Suppose that SK : X → X is well defined andcontinuous (e.g. X = Lp[0,1] with 1≤ p < ∞). The following statements are equivalent:
(i) There exist g, f ∈ L∞[0,1] such that
T (x)(s) = g(s)∫ s
0f (t)x(t)dt a.e.(s)
for all x ∈ X .
(ii) There exists h ∈ L∞[0,1] such that∫ 1
0T (x)(t)x ′(t)dt ≤
∫ 1
0
∣∣∣x(t)∫ 1
th(s)x ′(s)ds
∣∣∣dt
for all x ∈ X and x ′ ∈ X ′.
(iii) There exists h ∈ L∞[0,1] such that
|T ′(x ′)(t)| ≤∣∣∣∫ 1
th(s)x ′(s)ds
∣∣∣ a.e.(t)
for all x ′ ∈ X ′. E. Sanchez Factorization of operators on Banach function spaces
![Page 80: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/80.jpg)
Example
Consider the measure space given by the interval [0,1], its Borel σ -algebra and theLebesgue measure. Let K be the kernel given by K (s, t) = χ[0,s](t) for all s, t ∈ [0,1].Then, SK is just the Volterra operator. Suppose that SK : X → X is well defined andcontinuous (e.g. X = Lp[0,1] with 1≤ p < ∞). The following statements are equivalent:
(i) There exist g, f ∈ L∞[0,1] such that
T (x)(s) = g(s)∫ s
0f (t)x(t)dt a.e.(s)
for all x ∈ X .
(ii) There exists h ∈ L∞[0,1] such that∫ 1
0T (x)(t)x ′(t)dt ≤
∫ 1
0
∣∣∣x(t)∫ 1
th(s)x ′(s)ds
∣∣∣dt
for all x ∈ X and x ′ ∈ X ′.
(iii) There exists h ∈ L∞[0,1] such that
|T ′(x ′)(t)| ≤∣∣∣∫ 1
th(s)x ′(s)ds
∣∣∣ a.e.(t)
for all x ′ ∈ X ′. E. Sanchez Factorization of operators on Banach function spaces
![Page 81: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/81.jpg)
Example
Consider now the Hardy operator which is given by the kernel K (s, t) = 1s χ[0,s](t) for
all s, t ∈ [0,1]. Suppose that SK : X → X is well defined and continuous (e.g.X = Lp[0,1] with 1 < p < ∞). The following statements are equivalent:
(i) There exist g, f ∈ L∞[0,1] such that
T (x)(s) =g(s)
s
∫ s
0f (t)x(t)dt a.e.(s)
for all x ∈ X .
(ii) There exists h ∈ L∞[0,1] such that∫ 1
0T (x)(t)x ′(t)dt ≤
∫ 1
0
∣∣∣x(t)∫ 1
t
h(s)x ′(s)
sds∣∣∣dt
for all x ∈ X and x ′ ∈ X ′.
(iii) There exists h ∈ L∞[0,1] such that
|T ′(x ′)(t)| ≤∣∣∣∫ 1
t
h(s)x ′(s)
sds∣∣∣ a.e.(t)
for all x ′ ∈ X ′.
E. Sanchez Factorization of operators on Banach function spaces
![Page 82: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/82.jpg)
Example
Consider now the Hardy operator which is given by the kernel K (s, t) = 1s χ[0,s](t) for
all s, t ∈ [0,1]. Suppose that SK : X → X is well defined and continuous (e.g.X = Lp[0,1] with 1 < p < ∞). The following statements are equivalent:
(i) There exist g, f ∈ L∞[0,1] such that
T (x)(s) =g(s)
s
∫ s
0f (t)x(t)dt a.e.(s)
for all x ∈ X .
(ii) There exists h ∈ L∞[0,1] such that∫ 1
0T (x)(t)x ′(t)dt ≤
∫ 1
0
∣∣∣x(t)∫ 1
t
h(s)x ′(s)
sds∣∣∣dt
for all x ∈ X and x ′ ∈ X ′.
(iii) There exists h ∈ L∞[0,1] such that
|T ′(x ′)(t)| ≤∣∣∣∫ 1
t
h(s)x ′(s)
sds∣∣∣ a.e.(t)
for all x ′ ∈ X ′.
E. Sanchez Factorization of operators on Banach function spaces
![Page 83: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/83.jpg)
Example
Consider now the Hardy operator which is given by the kernel K (s, t) = 1s χ[0,s](t) for
all s, t ∈ [0,1]. Suppose that SK : X → X is well defined and continuous (e.g.X = Lp[0,1] with 1 < p < ∞). The following statements are equivalent:
(i) There exist g, f ∈ L∞[0,1] such that
T (x)(s) =g(s)
s
∫ s
0f (t)x(t)dt a.e.(s)
for all x ∈ X .
(ii) There exists h ∈ L∞[0,1] such that∫ 1
0T (x)(t)x ′(t)dt ≤
∫ 1
0
∣∣∣x(t)∫ 1
t
h(s)x ′(s)
sds∣∣∣dt
for all x ∈ X and x ′ ∈ X ′.
(iii) There exists h ∈ L∞[0,1] such that
|T ′(x ′)(t)| ≤∣∣∣∫ 1
t
h(s)x ′(s)
sds∣∣∣ a.e.(t)
for all x ′ ∈ X ′.
E. Sanchez Factorization of operators on Banach function spaces
![Page 84: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/84.jpg)
Example
Consider now the Hardy operator which is given by the kernel K (s, t) = 1s χ[0,s](t) for
all s, t ∈ [0,1]. Suppose that SK : X → X is well defined and continuous (e.g.X = Lp[0,1] with 1 < p < ∞). The following statements are equivalent:
(i) There exist g, f ∈ L∞[0,1] such that
T (x)(s) =g(s)
s
∫ s
0f (t)x(t)dt a.e.(s)
for all x ∈ X .
(ii) There exists h ∈ L∞[0,1] such that∫ 1
0T (x)(t)x ′(t)dt ≤
∫ 1
0
∣∣∣x(t)∫ 1
t
h(s)x ′(s)
sds∣∣∣dt
for all x ∈ X and x ′ ∈ X ′.
(iii) There exists h ∈ L∞[0,1] such that
|T ′(x ′)(t)| ≤∣∣∣∫ 1
t
h(s)x ′(s)
sds∣∣∣ a.e.(t)
for all x ′ ∈ X ′.
E. Sanchez Factorization of operators on Banach function spaces
![Page 85: The Radon-Nikody ́m Theorem for vector measures and factorization of operators on Banach function spaces](https://reader030.vdocuments.site/reader030/viewer/2022032714/55abf82d1a28aba6528b46d7/html5/thumbnails/85.jpg)
Example
Consider now the Hardy operator which is given by the kernel K (s, t) = 1s χ[0,s](t) for
all s, t ∈ [0,1]. Suppose that SK : X → X is well defined and continuous (e.g.X = Lp[0,1] with 1 < p < ∞). The following statements are equivalent:
(i) There exist g, f ∈ L∞[0,1] such that
T (x)(s) =g(s)
s
∫ s
0f (t)x(t)dt a.e.(s)
for all x ∈ X .
(ii) There exists h ∈ L∞[0,1] such that∫ 1
0T (x)(t)x ′(t)dt ≤
∫ 1
0
∣∣∣x(t)∫ 1
t
h(s)x ′(s)
sds∣∣∣dt
for all x ∈ X and x ′ ∈ X ′.
(iii) There exists h ∈ L∞[0,1] such that
|T ′(x ′)(t)| ≤∣∣∣∫ 1
t
h(s)x ′(s)
sds∣∣∣ a.e.(t)
for all x ′ ∈ X ′.
E. Sanchez Factorization of operators on Banach function spaces